Finite Group Databasebeta

← #37545301

Subgroups — #37545301

Order 23168.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153845C211yesmaximal, characteristic, complemented
2C2×#19153837C211yesmaximal, characteristic, complemented
3#26041949C211yesmaximal, characteristic, complemented
4#26041941C211yesmaximal, characteristic, complemented
5#26041666C211yesmaximal, characteristic, complemented
6#26041531C211yesmaximal, characteristic, complemented
7#26041507C211yesmaximal, characteristic, complemented
8#26041502C211yesmaximal, characteristic, complemented
9#26041442C211yesmaximal, characteristic, complemented
10#26041425C211yesmaximal, characteristic, complemented
11#26041423C211yesmaximal, characteristic, complemented
12#26041416C211yesmaximal, characteristic, complemented
13#26041384C211yesmaximal, characteristic, complemented
14#26041349C211yesmaximal, characteristic, complemented
15#460×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668448C2211yescharacteristic
18C2×#15668448C2211yescharacteristic
19C2×Q8×D181C2211yescharacteristic
20#19153845—24no—
21#19153837—24no—
22C2×Dic724C2211yescharacteristic
23C2×#15668439C2211yescharacteristic
24C2×#15668438C2211yescharacteristic
25C2×#15668437C2211yescharacteristic
26C2×#15668437C2211yescharacteristic
27C2×#15668434C2211yescharacteristic
28C2×#15668434C2211yescharacteristic
29C2×#15668432C2211yescharacteristic
30C2×C8×D181C2211yescharacteristic
31#19153812—21no—
32#19153804—21no—
33#19153804C2211yescharacteristic
34#19153796—21no—
35#19153796C2211yescharacteristic
36#19153791—21no—
37#19153791C2211yescharacteristic
38#19153788—21no—
39#19153788—21no—
40#19153787C2211yescharacteristic
41#19153787C2211yescharacteristic
42C4×Dic362C2211yescharacteristic
43C4×D724—21no—
44C4×D724C2211yescharacteristic
45D4×Dic181—21no—
46#19153747C2211yescharacteristic
47#19153741C2211yescharacteristic
48#19153740C2211yescharacteristic
49#19153739C2211yescharacteristic
50#19153738C2211yescharacteristic
51#19153728C2211yescharacteristic
52#19153728C2211yescharacteristic
53#19153727C2211yescharacteristic
54C2×SD16×C181C2211yescharacteristic
55C2×SD16×C181C2211yescharacteristic
56#127×C181C2211yescharacteristic
57C4×Q8×C181C2211yescharacteristic
58C4⋊C8×C181C2211yescharacteristic
59#102×C181C2211yescharacteristic
60#102×C181C2211yescharacteristic
61#15668450—22no—
62#15668450—41no—
63#15668448—22no—
64#15668448—22no—
65#15668448—41no—
66#15668448—41no—
67C2×Dic362—21no—
68C2×Dic362C2311yescharacteristic
69C2×Dic362C2311yescharacteristic
70Q8×D181—22no—
71Q8×D181—41no—
72C2×D724—21no—
73C2×D724—21no—
74C2×D724D411yescharacteristic
75C2×D724C2311yescharacteristic
76C2×C22⋊D181—21no—
77C2×C22⋊D181—21no—
78C2×C22⋊D181—21no—
79C2×C22⋊D181—21no—
80C22×Dic181—21no—
81C22×Dic181—21no—
82C2×C4×D181—21no—
83C2×C4×D181—21no—
84C2×C4×D181C2311yescharacteristic
85C2×C4×D181C2311yescharacteristic
86Dic724—22no—
87#15668439D414yescomplemented
88#15668438—22no—
89#15668437—22no—
90#15668437—41no—
91#15668434—22no—
92#15668434—41no—
93#15668432—22no—
94C2×#13089678C2311yescharacteristic
95C2×#13089678C2311yescharacteristic
96C8×D181—22no—
97C4⋊Dic181—21no—
98C4⋊Dic181D411yescharacteristic
99C4⋊Dic181C2311yescharacteristic
100#15668428—21no—
101C22⋊Dic181—21no—
102C22⋊Dic181—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426D411yescharacteristic
108#15668426D411yescharacteristic
109C4×Dic181—21no—
110C2×Q8×C181C2311yescharacteristic
111C2×D4×C181—21no—
112C2×D4×C181—21no—
113C2×D4×C181C2311yescharacteristic
114C2×D4×C181C2311yescharacteristic
115SD16×C181—22no—
116SD16×C181—22no—
117C2×C8×C181C2311yescharacteristic
118C2×C8×C181C2311yescharacteristic
119C4⋊C4×C181—21no—
120C4⋊C4×C181C2311yescharacteristic
121C42×C181—21no—
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181C2×D411yescharacteristic
125C22×D181C2×D411yescharacteristic
126Dic362—21no—
127Dic362—21no—
128Dic362—21no—
129Dic362—42no—
130Dic362C2×D412yes—
131D724—21no—
132D724—21no—
133D724—41no—
134D724—42no—
135D724—42no—
136C22⋊D181—42no—
137C22⋊D181—42no—
138C22⋊D181—81no—
139C22⋊D181—81no—
140C2×Dic181—21no—
141C2×Dic181—21no—
142C2×Dic181—21no—
143C2×Dic181—42no—
144C2×Dic181—42no—
145C2×Dic181C2×D411yescharacteristic
146C2×Dic181C2×D411yescharacteristic
147C4×D181—21no—
148C4×D181—21no—
149C4×D181—21no—
150C4×D181—21no—
151C4×D181—42no—
152C4×D181—42no—
153#13089678—21no—
154#13089678C2×D412yes—
155C23×C181—21no—
156C23×C181—21no—
157Q8×C181—21no—
158Q8×C181C2×D412yes—
159D4×C181—21no—
160D4×C181—21no—
161D4×C181—21no—
162D4×C181—21no—
163D4×C181—42no—
164D4×C181—81no—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2411yescharacteristic
170C8×C181—21no—
171C8×C181—21no—
172C22×C181—41no—
173C22×C181—41no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181—41no—
178C22×C181C22×D411yescharacteristic
179C2×D181—21no—
180C2×D181—21no—
181C2×D181—21no—
182C2×D181—21no—
183C2×D181—21no—
184C2×D181—21no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—41no—
188C4×C181—22no—
189C4×C181—21no—
190C4×C181—21no—
191C4×C181—41no—
192C4×C181C22×D411yescharacteristic
193C4×C181H(5,𝔽2)11yescharacteristic
194Dic181—21no—
195Dic181—21no—
196Dic181—22no—
197Dic181—41no—
198Dic181—41no—
199C2×C181—81no—
200C2×C181—81no—
201C2×C181D4211yescharacteristic
202C2×C181#57511yescharacteristic
203C2×C181#57711yescharacteristic
204D181—41no—
205D181—41no—
206D181—81no—
207C181#327111yescharacteristic, complemented, Sylow
208#3271—1811nomaximal, Sylow
209C2×#137—1811no—
210C2×#135—1811no—
211#550—1811no—
212#548—1811no—
213#496—1811no—
214#469—1811no—
215#465—1811no—
216#464—1811no—
217#460D18111yescharacteristic, complemented
218#452—1811no—
219#449—1811no—
220#449—1811no—
221#448—1811no—
222#441—1811no—
223#435—1811no—
224C2×#41—1811no—
225C2×#41—1811no—
226C2×#41—1811no—
227C22×Q8—1811no—
228#137—3624no—
229#135—3624no—
230C2×Q16—1811no—
231C2×Q16—1811no—
232C2×SD16—1811no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16C2×D18111yescharacteristic
236C2×SD16C2×D18111yescharacteristic
237C2×D8—1811no—
238C2×D8—1811no—
239C2×M16—1811no—
240C22×C8—1811no—
241#127—3621no—
242#127C2×D18111yescharacteristic
243#124—3621no—
244#124—1811no—
245C22⋊Q8—3621no—
246C22⋊Q8—1811no—
247#121—3621no—
248#121—3621no—
249#121—3621no—
250#121—1811no—
251#121—1811no—
252#121—1811no—
253C4×Q8—1811no—
254C4×Q8C2×D18111yescharacteristic
255C4×D4—3621no—
256C4×D4—3621no—
257C4×D4—1811no—
258#106—1811no—
259C4⋊C8C2×D18111yescharacteristic
260#103—1811no—
261#103—1811no—
262#103—1811no—
263#102—1811no—
264#102C2×D18111yescharacteristic
265#102C2×D18111yescharacteristic
266C22⋊C8—1811no—
267C22⋊C8—1811no—
268#97—1811no—
269#41—3622no—
270#41—3622no—
271#41—3622no—
272#41—7241no—
273#41—7241no—
274#41—7241no—
275C2×Q8—3621no—
276C2×Q8—3622no—
277C2×Q8—1811no—
278C2×Q8—1811no—
279C2×Q8—7241no—
280C2×Q8C22×D18111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—3621no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—3621no—
288C2×D4—3621no—
289C2×D4—1811no—
290C2×D4—1811no—
291C2×D4C22×D18111yescharacteristic
292C2×D4C22×D18111yescharacteristic
293C22×C4—3621no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4—3621no—
297C22×C4—1811no—
298C22×C4—1811no—
299Q16—3622no—
300Q16—1814no—
301SD16—22no—
302SD16—22no—
303SD16—3622no—
304SD16—3622no—
305SD16—7241no—
306D8—3622no—
307D8—7241no—
308M16—3622no—
309C2×C8—3622no—
310C2×C8—1811no—
311C2×C8—1811no—
312C2×C8C22×D18111yescharacteristic
313C2×C8C22×D18111yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—3621no—
316C4⋊C4—3621no—
317C4⋊C4—1811no—
318C4⋊C4—1811no—
319C4⋊C4C22×D18111yescharacteristic
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—1811no—
327C22⋊C4—1811no—
328C42—21no—
329C42—3621no—
330C42C22×D18111yescharacteristic
331C23—21no—
332C23—21no—
333C23—3621no—
334C23—1811no—
335C23—1811no—
336Q8—21no—
337Q8—3621no—
338Q8—3621no—
339Q8—3621no—
340Q8—1812no—
341Q8—7242no—
342Q8D4×D18112yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—42no—
348D4—81no—
349D4—3621no—
350D4—3621no—
351D4—14481no—
352D4—14481no—
353D4—7241no—
354D4—7242no—
355D4—7242no—
356D4—7242no—
357D4—7242no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—3621no—
361C2×C4—3621no—
362C2×C4—3621no—
363C2×C4—3621no—
364C2×C4—3621no—
365C2×C4—3621no—
366C2×C4—3621no—
367C2×C4—1811no—
368C2×C4—1811no—
369C2×C4—7242no—
370C2×C4—7242no—
371C2×C4—7242no—
372C2×C4—7242no—
373C2×C4D4×D18111yescharacteristic
374C2×C4D4×D18111yescharacteristic
375C2×C4C23×D18111yescharacteristic
376C8—21no—
377C8—21no—
378C8—3621no—
379C8—1812no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—3621no—
387C22—3621no—
388C22—3621no—
389C22—3621no—
390C22—3621no—
391C22—3621no—
392C22—7241no—
393C22—7241no—
394C22—7241no—
395C22C2×D4×D18111yescharacteristic
396C4—22no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—3621no—
401C4—3621no—
402C4—3622no—
403C4—7241no—
404C4—7241no—
405C4C2×D4×D18111yescharacteristic
406C4#1915386211yescharacteristic
407C2—81no—
408C2—81no—
409C2—14481no—
410C2—7241no—
411C2—7241no—
412C2#2604191311yescharacteristic
413C2#2604207511yescharacteristic
414C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#460×C181C2
socleC22×C181C22×D4
hypercenter#460D181
hypocenterC181#3271
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19153845C2semidirect
2. C2×#19153837C2semidirect
3. #26041949C2semidirect
4. #26041941C2semidirect
5. #26041666C2semidirect
6. #26041531C2semidirect
7. #26041507C2semidirect
8. #26041502C2semidirect
9. #26041442C2semidirect
10. #26041425C2semidirect
11. #26041423C2semidirect
12. #26041416C2semidirect
13. #26041384C2semidirect
14. #26041349C2semidirect
15. #460×C181C2semidirect
87. #15668439D4semidirect
207. C181#3271semidirect
217. #460D181semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#191538452
C2×#191538372
#260419492
#260419412
#260416662
#260415312
#260415072
#260415022
#260414422
#260414252
#260414232
#260414162
#260413842
#260413492
#460×C1812
#3271181